Double cut - A pair of cuts with nothing between them may be drawn around any subgraph. Likewise two nested cuts with nothing between them may be erased. This rule is equivalent to Boolean involution.

Iteration/Deiteration – To understand this rule, it is best to view a graph as a tree structure having nodes and ancestors. Any subgraph P in node n may be copied into any node depending on n. Likewise, any subgraph P in node n may be erased if there exists a copy of P in some node ancestral to n (i.e., some node on which n depends). For an equivalent rule in an algebraic context, see C2 in Laws of form.

A proof manipulates a graph by a series of steps, with each step justified by one of the above rules. If a graph can be reduced by steps to the blank page or an empty cut, it is what is now called a tautology (or the complement thereof). Graphs that cannot be simplified beyond a certain point are analogues of the satisfiableformulas of first-order logic.

Peirce notated predicates using intuitive English phrases; the standard notation of contemporary logic, capital Latin letters, may also be employed. A dot asserts the existence of some individual in the domain of discourse. Multiple instances of the same object are linked by a line, called the "line of identity". There are no literal variables or quantifiers in the sense of first-order logic. A line of identity connecting two or more predicates can be read as asserting that the predicates share a common variable. The presence of lines of identity requires modifying the alpha rules of Equivalence.

The beta graphs can be read as a system in which all formula are to be taken as closed, because all variables are implicitly quantified. If the "shallowest" part of a line of identity has even (odd) depth, the associated variable is tacitly existentially (universally) quantified.

Add to the syntax of alpha a second kind of simple closed curve, written using a dashed rather than a solid line. Peirce proposed rules for this second style of cut, which can be read as the primitive unary operator of modal logic.

Zeman (1964) was the first to note that straightforward emendations of the gamma graph rules yield the well-known modal logics S4 and S5. Hence the gamma graphs can be read as a peculiar form of normal modal logic. This finding of Zeman's has gone unremarked to this day, but we included it in Wikipedia anyway.

But Peirce's evolving semiotic theory led him to doubt the value of logic formulated using conventional linear notation, and to prefer that logic and mathematics be notated in two (or even three) dimensions. His work went beyond Euler's diagrams and Venn's revision thereof. Frege's 1879 Begriffsschrift also employed a two-dimensional notation for logic, but one very different from Peirce's.

Peirce's first published paper on graphical logic (reprinted in Vol. 3 of his Collected Papers) proposed a system dual (in effect) to the alpha existential graphs, called the entitative graphs. He very soon abandoned this formalism in favor of the existential graphs. The graphical logic went unremarked during his lifetime, and was invariably denigrated or ignored after his death, until the Ph.D. theses by Roberts (1964) and Zeman (1964).

Currently, the chronological critical edition of Peirce's works, the Writings, extends only to 1892. Much of Peirce's work on logical graphs consists of manuscripts written after that date and still unpublished. Hence our understanding of Peirce's graphical logic is likely to change as the remaining 23 volumes of the chronological edition appear.